
// Make newform 648.2.n.l in Magma, downloaded from the LMFDB on 28 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_648_n();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_648_n_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_648_2_n_l();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_648_2_n_l();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [1, 0, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_648_n();"
function MakeCharacter_648_n()
    N := 648;
    order := 6;
    char_gens := [487, 325, 569];
    v := [6, 3, 2];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_648_n_Hecke();"
function MakeCharacter_648_n_Hecke( : Kf := false)
    N := 648;
    order := 6;
    char_gens := [487, 325, 569];
    char_values := [[1, 0, 0, 0], [-1, 0, 0, 0], [0, 0, -1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..prec]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0, 1, 1, -1], [0, 0, 0, 0], [2, -2, -1, 2], [-1, -1, 1, 2], [-1, 3, -1, 0], [4, 1, -2, -1], [-2, 2, 0, -1], [-1, 0, 2, 5], [0, 1, -7, 1], [3, 0, 3, 0], [0, 0, -2, 0], [0, 0, 0, -5], [0, -2, -8, -2], [5, -1, 5, 0], [0, 2, 0, -4], [4, 0, -8, -6], [12, 2, -6, -2], [0, -9, 0, 0], [-2, 3, 1, -3], [-3, -10, 0, 5], [9, 0, 0, 0], [-7, -1, 7, 2], [0, -14, 0, 0], [-10, 2, 0, -1], [8, 0, -8, 0], [0, 8, 0, 0], [0, -2, 0, -2], [-6, 0, 12, 0], [8, 0, -16, -1], [0, -1, 6, -1], [-5, 10, 0, -5], [6, 5, -3, -5], [12, -3, -12, 6], [4, -16, -2, 16], [-6, 0, 3, 0], [0, 8, 0, -16], [4, 1, -2, -1], [-2, 0, 4, 10], [0, -1, 13, -1], [-1, 0, -1, 0], [0, 0, 0, 14], [2, 0, -4, -10], [-1, 9, 1, -18], [0, 8, 3, 8], [-7, 0, 14, 8], [-8, -12, 0, 6], [-22, 7, 11, -7], [-11, 7, 11, -14], [-1, 5, -1, 0], [12, -1, -6, 1], [2, -2, 0, 1], [0, 8, 6, 8], [1, -2, -1, 4], [1, 0, -2, -9], [0, 5, 12, 5], [-8, -14, 8, 28], [-1, 0, 2, -24], [9, -2, 0, 1], [8, 4, 8, 0], [-12, 1, 12, -2], [12, 4, -6, -4], [-10, -12, 5, 12], [12, 0, -24, -6], [0, -10, 2, -10], [1, 2, -1, -4], [-3, -4, -3, 0], [7, -1, 7, 0], [0, 0, -4, 0], [2, -5, -1, 5], [8, 2, 8, 0], [0, -4, 0, 8], [-11, -14, 0, 7], [-6, -2, 6, 4], [8, -16, -4, 16], [0, 0, 0, -10], [0, -3, 15, -3], [-4, 10, -4, 0], [2, 0, -4, 29], [0, -9, -6, -9], [0, 16, -1, 16], [4, -18, -2, 18], [0, 13, 0, 0], [6, 32, 0, -16], [-1, 24, 0, -12], [30, 0, -30, 0], [10, -12, 10, 0], [-18, 8, 0, -4], [-11, -10, 11, 20], [-10, 24, -10, 0], [0, 2, -10, 2], [12, 0, -24, 4], [7, 5, -7, -10], [2, 32, 0, -16], [-6, -19, 3, 19], [-6, 3, 3, -3], [-18, -20, 0, 10], [-20, 12, 10, -12], [6, -24, 0, 12], [1, 0, -2, -15], [-18, 0, 36, 7], [0, -26, 0, 0], [-11, 0, 22, 6], [28, 0, -14, 0], [18, -9, -18, 18], [-28, 2, 14, -2], [-21, 28, 0, -14], [25, 1, 25, 0], [-30, -18, 0, 9], [0, -14, -16, -14], [7, 14, -7, -28], [0, -19, -7, -19], [6, 0, -12, 14], [0, -7, 14, -7], [-14, -18, -14, 0], [-20, -24, 0, 12], [-4, 11, 4, -22], [12, 4, -6, -4], [3, -54, 0, 27], [48, 6, -24, -6], [-11, -9, -11, 0], [-36, 1, 18, -1], [13, -14, -13, 28], [10, -12, 10, 0], [22, 0, -44, -4], [1, -11, 1, 0], [7, 0, -14, -2], [-6, 37, -6, 0], [-29, 22, 0, -11], [27, -13, -27, 26], [4, -26, -2, 26], [6, 0, -12, -20], [0, -12, 10, -12], [0, -13, -5, -13], [-14, 0, 28, 6], [0, 15, -10, 15], [0, -6, 29, -6], [-5, 0, 10, -32], [34, 15, -17, -15], [2, 14, -1, -14], [-30, 22, 0, -11], [12, 0, -24, 28], [9, 14, 9, 0], [0, 6, 28, 6], [-5, 0, 10, 21], [-18, 0, 36, -22], [-47, -3, 47, 6], [-4, -12, -4, 0], [12, 7, -12, -14], [0, -2, 0, 2], [-3, 26, 0, -13], [44, -19, -22, 19], [14, -16, 0, 8], [-10, 0, 20, 16], [0, 3, 3, 3], [-6, 22, -6, 0], [-15, 1, 15, -2], [-9, -30, 0, 15], [-22, -13, 22, 26], [17, 28, 0, -14], [-42, -18, 21, 18], [3, -15, 3, 0], [18, -10, 0, 5], [0, 3, -5, 3], [7, 0, -14, 39], [0, -12, 2, -12], [-2, -2, 2, 4], [-13, -18, 0, 9], [4, -21, 4, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_648_n_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_648_2_n_l();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_648_2_n_l(:prec:=4)
    chi := MakeCharacter_648_n();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_648_2_n_l();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_648_2_n_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_648_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 6, 11, -6, 1]>,<13,R![121, -132, 59, -12, 1]>,<17,R![1, 4, 1]>],Snew);
    return Vf;
end function;